weak convergence in Sobolev spaces and pointwise convergence?

Question

I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that

$\|u_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u_m)\rightharpoonup I$ in $L^p(\mathbb{R}^n)$,

and then it assume in addition that

$u_m\rightharpoonup u$ weakly in $H^{1,2}(\mathbb{R}^n)$ and pointwise almost everywhere.

My question is

why the pointwise convergence assumption is reasonable? Since $\mathbb R^n$ is not compact, the embedding theorem is not obviously valid.

Thanks in advance.

Answer 1

Just this: for all $r > 0$ the restriction of $u_m$ to the ball $B_r$ is compact in $L^1({B_r})$, hence it has a subsequence converging a.e. there. By a standard diagonal argument, there is a single subsequence of $u_m$ converging a.e. in $\mathbb{R}^n$.